The light distribution in the disks of many galaxies is non-axisymmetric or `lopsided' with a spatial extent much larger along one half of a galaxy than the other, as in M101. Recent near-IR observations show that lopsidedness is common. The stellar disks in nearly 30 % of galaxies have significant lopsidedness, greater than 10 % measured as the Fourier amplitude of the m=1 component normalized to the average value. This asymmetry is traced particularly well by the atomic hydrogen gas distribution lying in the outer parts. The lopsidedness also occurs in the nuclear regions, where the nucleus is offset with respect to the outer isophotes. The galaxies in a group environment show higher lopsidedness. The origin of lopsidedness could be due to the disk response to a tidally distorted halo, or via gas accretion. The lopsidedness has a large impact on the dynamics of the galaxy, its evolution, the star formation in it, and on the growth of the central black hole and on the nuclear fueling, merging of binary black holes etc. The disk lopsidedness can be used as a diagnostic to study the halo asymmetry. This is an emerging area in galactic structure and dynamics. In this review, the observations to measure the lopsided distribution, as well as the theoretical progress made so far to understand its origin and properties, and the related open problems will be discussed. (abridged).

75 pages, 28 figures, a review article, to be published by Physics Reports